MONOPOLISTIC COMPETITION WHEN INCOME
MATTERS*
Paolo Bertoletti and Federico Etro
We analyse monopolistic competition when consumers have an indirect utility that is additively separable. This leads to markups depending on income (both in the short and long run) but not on the market size, which generates pricing to market, incomplete pass-through and pure gains from variety for countries that open up to trade. Firms’ heterogeneitya la Melitz implies a Darwinian effect of consumers’ spending on business creation and a Linderian effect on (endogenous) quality provision. We discuss extensions with an outside good and heterogenous agents, and offer simple and tractable specifications (linear or log-linear) of the demand functions.
This article proposes an alternative model of monopolistic competition in the tradition of the studies of large markets (Chamberlin, 1933), where firms choose prices independently and entry is free. The model is based on a class of non-homothetic preferences, unexplored in the analysis of monopolistic competition, which satisfy indirect additivity and delivers convenient specifications for applied research, especially in trade and macroeconomics.
As is well known, the model introduced by Dixit and Stiglitz (D–S, 1977: Section I) and based on constant-elasticity-of-substitution (CES) preferences over differentiated goods has become a workhorse model in modern economics. It implies constant markups and an endogenous number of firms that is proportional to both the number of consumers and their per capita income. Moreover, under firm heterogeneity, CES preferences imply that changes in population or income do not affect the efficiency of the active firms. These features have key consequences, for instance on the structure of and the gains from trade (Krugman, 1980; Melitz, 2003) and on firms’ behaviour over the business cycle (see Blanchard and Kiyotaki, 1987 and, more recently, Bilbiie et al., 2012).
From an empirical point of view, however, the CES model has some drawbacks. Primarily, it cannot account for the variability of markups across countries and/or over the business cycle. There is indeed a consistent evidence in the trade literature that, over the long run, prices and markups tend to be higher in richer countries (Alessandria and Kaboski, 2011; Fieler, 2011); there is also some macroeconomic evidence that markups are variable over the business cycle (for instance, Nekarda and Ramey (2013) make a case for procyclical markup behaviour in reaction to demand shocks).1Moreover, although
* Corresponding author: Paolo Bertoletti, Department of Economics and Management, University of Pavia, Via San Felice, 5, I-27100 Pavia, Italy. Email: paolo.bertoletti@unipv.it.
The authors are grateful to the Editor Martin Cripps, Daron Acemoglu, Simon Anderson, Paolo Epifani, Gene Grossman, Atsushi Kajii, James Markusen, Volker Nocke, Patrick Rey, Yossi Spiegel, Jacques Thisse, Kresimir Zigic and many anonymous referees. Seminar participants at the 2013 Cresse Conference in Corfu, the HSE Center for Market Studies and Spatial Economics in St. Petersburg, the University of Pavia, Ca’ Foscari University, Kyoto University, Hitotsubashi University and Yonsei University (Seoul) provided insightful comments.
1 There is also evidence on incomplete pass-through for changes of marginal costs and trade costs (De
Loecker et al., 2012).
the empirical analysis of the impact of market size on prices under monopolistic competition has rarely distinguished between income and population effects, Simonovska (2015) studies international pricing of traded goods (online sales of clothes shipped to foreign markets and in competition with many imperfect substitutes) and finds a positive elasticity of prices with respect to per capita income but no significant impact of population on prices.
To account for the variability of markups under monopolistic competition, one has to depart from CES preferences. But models based on quasilinear (Spence, 1976; Melitz and Ottaviano, 2008; Anderson et al., 2012) or general homothetic preferences (Benassy, 1996) face similar limitations, since they remove any direct effect of income on demand (with quasilinearity) or on demand elasticity (with homotheticity). To obtain variable markups, D–S (1977: Section II) proposed non-homothetic preferences represented by additively separable direct utility functions (Krugman, 1979). As recently stressed, such ‘direct additivity’ generates equilibrium prices that can either decrease or increase in the number of consumers (Zhelobodko et al., 2012), implying an ambiguous impact of market size on welfare and ambiguous selection effects under firm heterogeneity (Bertoletti and Epifani, 2014; Dhingra and Morrow, 2015). However, in spite of non-homotheticity, free entry neutralises the impact of income on markups and market structure: in the long-run equilibrium, when the number of firms adjusts to obtain zero expected profits, prices and firm selection cannot be affected by changes in consumers’ expenditure (Zhelobodko et al., 2012; Bertoletti and Etro, 2014a, b). Notice that on the one hand monopolistic competition models a la D–S (1977) were developed to explain the number of varieties/competitors in the long run, on the other hand the empirical relation mentioned between markups and income emerging in the trade literature (Alexandria and Kaboski, 2011; Fieler, 2011; Simonovska, 2015) refers to a time span that explicitly requires a long-run theoretical foundation. We propose an alternative class of non-homothetic preferences and argue that it can easily account for the stylised facts outlined above and, in particular, for the relation between prices and income in the long run.2
Nevertheless, it is important to remark that, for a given number of firms, income affects markups also under direct additivity of preferences (Behrens and Murata, 2012a). Moreover, there exist other models based on non-homothetic preferences in which income heterogeneity between individuals or across countries ‘matter’ in the sense of affecting crucially the market results even in the long run. For instance, Simonovska (2015) studies a multi-country trade model where wages are endogenous and exert an impact on the equilibrium markups, while Behrens and Murata (2012b) show that differences in labour productivity affect the relative gains from trade. In another important work based on non-homothetic prefer-ences, Murata (2009) exploits increasing returns due to specialisation arising out of differentiated intermediate goods. This implies that higher labour efficiency
2 On the crucial role of non-homotheticity in trade models see also the recent works by Fajgelbaum et al.
affects prices (and the number and composition of varieties) even in the long run because it allows for more specialisation and lower marginal costs.3 Our complementary approach in this article is to present a simple model in the D–S tradition where income affects markups in the long run even in autarky, without heterogeneous firms/varieties or consumers and without returns from specialisa-tion.
Our main assumption is that consumers’ preferences can be represented by an additively separable indirect utility function. Such ‘indirect additivity’ amounts to assume that the relative demand of two goods does not depend on the price of other goods, while it depends in general on income. It is thus different from ‘direct additivity’, for which the marginal rate of substitution between any two goods does not depend on the consumption of other goods. In fact, duality theory (Hicks, 1969; Samuelson, 1969) tells us that direct and indirect additivity characterise two different classes of well-behaved preferences (remarkably, the homothetic case of CES preferences is the only common ground). A key implication of our assumption is that the number of goods provided in the market does not affect their substitutability and thus the demand elasticity, while income can affect both with crucial conse-quences. Remarkably, simple and common direct demand functions, such as linear demands and log-linear demands, emerge from our preferences (with ‘addilog’ and exponential subutilities) and generate simple closed form solutions for variable markups, which can be easily used in a variety of applied situations.
Monopolistic competition under indirect additivity produces two-sided results that can be useful for trade and macroeconomic applications. First, it generalises the neutrality of market size on the production structure which emerges in CES models, thereby yielding pure gains from varieties as in Krugman (1980). Second, it delivers markups that are variable in income/spending, generating pricing to market as a natural phenomenon: as long as demand is less elastic for richer consumers, markups are higher in markets with higher individual income. Moreover, markups do change when demand shocks affect individual spending or supply shocks affect marginal costs. We show that these results are robust to extensions in which an outside good represents the rest of the economy, there is consumer heterogeneity in both preferences and income, and also when firms differ in productivity a la Melitz (2003). At the very least, our results suggest that empirical works should carefully distinguish the effects due to per capita income from those associated to the population size.
The comparative statics for business creation is also of interest. Richer consumers with less elastic demand induce firms to increase their markups, which triggers a more than proportional entry of firms, while an increase in income inequality between consumers tends to exert the opposite effects. Instead, when firms are heterogeneous in productivity, indirect additivity establishes a Darwinian mechanism that is absent in the Melitz model: less productive firms enter in booms (when income increases) and exit during downturns (a sort of ‘cleansing effect’ of recessions). Finally, if firms can invest in the quality of their products, we find that more productive firms tend to react
3 The article by Murata (2009) is akin to our own in stressing the different role played by population and
per capita income in affecting the elasticity of demand. It is different because it assumes directly additive preferences and asymmetric final products sold in competitive markets.
to an increase in consumers’ income by offering products of higher quality sold at higher prices, which is consistent with the celebrated Linderian effect (Linder, 1961). The work is organised as follows. In Section 1 we present our baseline model characterising the endogenous entry equilibrium and introducing convenient specifi-cations to be used in applied research. We also compare our results with those of alternative models based on directly additive, homothetic and non-separable prefer-ences. In Section 2 we extend the model in various directions and compare optimal and equilibrium market structures. In Section 3 we study a two-country modela la Krugman considering both costless trade between different countries and costly trade between identical countries. We conclude in Section 4. All the proofs are in the Appendix.
1. A Model of Monopolistic Competition
Consider a market populated by L identical agents with income E > 0 to be spent on a mass of n differentiated goods.4 We represent preferences through indirect utilities, which depend on the prices pj of each variety j and on income or, exploiting
homogeneity of degree zero of the indirect utility, on their ratios sj pj=E. Our key
assumption is the adoption of the following symmetric and additively separable indirect utilities: V ¼ Z n 0 v pj E dj: (1)
As we will clarify below, this assumption identifies a general class of well-behaved preferences that do not satisfy the D–S (1977) assumption of direct additivity, with the remarkable exception of the CES case. To satisfy sufficient conditions for (1) being a valid indirect utility function (while allowing for a possibly finite choke-off price and obtaining well-behaved demand functions), we assume that the indirect sub-utility v(s) is at least thrice differentiable, with v(s) > 0, v0ðsÞ \ 0 and v00ðsÞ [ 0 for any s \s, v(s) = 0 for s s, and lims!svðsÞ; v0ðsÞ ¼ 0 for some s [ 0. These assumptions imply
that demand and extra utility are zero for a good that is not consumed.
The Roy identity provides the following direct individual demand function for good i:
xiðpi; E; lÞ ¼ v0 pi E l ; (2) where l ¼ Z n 0 v0 pj E pj Edj (3)
depends on the marginal utility of income: namely, l = E(@V/@E) < 0. The demand function of each variety depends on its price and on the same price aggregator, l, which is unaffected by the price pj of a single firm. Total market demand is
qi ¼ xiðpi; E; lÞL.
4 Using the wage as numeraire, E can be interpreted as the labour endowment of each agent (in efficiency
Preferences represented by (1) are homothetic if and only if v(s) is isoelastic, i.e., if vðsÞ ¼ s1h with h> 1. Indeed, in such a case they are of the CES type, with indirect
utility V ¼ EðRjp1h
j djÞ1=ðh1Þ, where h is the elasticity of substitution. By an important
result in duality theory (Hicks, 1969; Samuelson, 1969) the class of preferences which satisfy ‘direct additivity’, i.e., that can be represented by an additive direct utility U ¼ R0nuðxjÞdj, for some well-behaved subutility u(), does not satisfy (1), with the
only exception of CES preferences.5Therefore, the indirect utility (1) encompasses an unexplored class of non-homothetic preferences whose corresponding direct utility functions are non-additive.
Suppose that each variety is sold by a firm producing with constant marginal cost c > 0 and fixed cost F > 0 (both in labour units): the profits of firm i can then be written as:
pðpi; E; lÞ ¼ ðpi cÞv 0 pi E L l F : (4)
The most relevant implication of (2) is that the elasticity of the direct demand corresponds to the (absolute value of the) elasticity of v0ðÞ, which we define as:
hðsÞ v00ðsÞs v0ðsÞ [0:
This elasticity depends on the price as a fraction of income, pi=E, but is independent
from l and L.6 Instead, in the case of direct additivity the elasticity of the inverse demand is uniquely determined by the consumption level.7
Any firm i maximises (4) with respect to pi. The first order condition (FOC) is
v0ðsiÞ þ ðpi cÞv00ðsiÞ=E ¼ 0, which requires that (locally) h(s) > 1. Moreover the
second order condition (SOC) requires 2h(s) > f(s), where fðsÞ v000ðsÞs=v00ðsÞ is a measure of demand curvature. Notice that h0ðsÞs=hðsÞ ¼ hðsÞ þ 1 fðsÞ, therefore h0[ 0 if and only if h > f 1, in which case demand becomes more elastic when price
goes up or income goes down.8The FOC for profit-maximising price can be rewritten as follows:9 pe c pe ¼ 1 h pe E : (5)
This pricing rule shows that under indirect additivity the profit maximising price is always independent from the mass of varieties supplied, because the latter does not
5 In addition, it can be shown that the assumptions of either direct or indirect additivity and homotheticity
imply that preferences are CES: see Blackorby et al. (1978, section 4.5.3).
6 Notice that h is a measure of the curvature of v(): as such, it could be related to well-known risk aversion
measures (Bertoletti, 2006; Behrens and Murata, 2007).
7 Under direct additivity, the (individual) inverse demand of variety i is given by p
iðxi; kÞ ¼ u0ðxiÞ=k, where
k ¼Rn
0uðxjÞxjdj=E is the marginal utility of income. Notice that each inverse demand depends on its own
quantity and on the same quantity aggregator, k. Accordingly, both direct and indirect additivity satisfy the so-called ‘generalized additive separability’: see Pollak (1972).
8 If demand is (locally) concave (v000[ 0) the SOC is always satisfied and h0[ 0. On the contrary, if
demand is convex (v000\ 0) we may have h0\ 0.
9 To guarantee the existence of a solution to (5) we assumesE [ c (the consumer willingness to pay is
affect the elasticity of demand. This property appears entirely consistent with the Chamberlinian treatment of the ‘large group equilibrium’ (Chamberlin, 1933, ch. V). It follows that an exogenous increase in the number of competitors would just proportionally reduce the level of individual consumption.10 At the same time, the optimal price grows with income if firms face a less elastic demand (and vice versa), which provides a demand-side rationale for markups that are variable across markets (or over the business cycle). Consider the realistic case of h0[ 0: then, the model is consistent with typical forms of pricing-to-market, i.e., the same good should be sold at a higher price in richer (or booming) markets.11Similarly, under the same assumption a change in the marginal cost is transmitted (pass-through) to prices in a less than proportional way (undershifting). Summing up, we have:
PROPOSITION 1. Under indirect additivity and monopolistic competition the
profit-maximising prices are independent from the mass of active firms; they increase in the income of consumers and less than proportionally in the marginal cost, if and only if demand elasticity is increasing in the price.
Since by symmetry the equilibrium profit is the same for all firms and it is decreasing in their mass, we can characterise the endogenous market structure through the zero profit condition (p c)EL/np = F. This and the pricing rule (5) jointly deliver the free-entry equilibrium mass of firms and production size of each firm:
ne ¼ EL F h p e E ; qe¼ F h pe E 1 c : (6)
The following Proposition summarises the comparative statics for ne:
PROPOSITION 2. Under indirect additivity, in a monopolistic competition equilibrium with
endogenous entry the mass of firms increases proportionally with the number of consumers; it increases more than proportionally with the income of consumers and decreases with the marginal cost if and only if the demand elasticity is increasing in the price.
As a corollary, the equilibrium size of each firm qe in (6) does not depend on the
number of consumers and it decreases with individual income if and only if h0 [ 0. To understand these results and their applications, it is convenient to think of changes in L as changes in the scale of the economy, of changes in E as (due to) demand shocks on the disposable income of consumers and of changes in c as supply shocks to firms’ productivity. First of all, the impact of an increase in the scale of the economy is always the same as under CES preferences: a larger market size does not affect prices and production per firm but simply attracts more firms without inducing any other effect on
10 In the D–S case of direct additivity, an exogenous increase in the number of varieties n could either
increase or decrease the price level, cutting more or less than proportionally the level of individual consumption.
the market structure. This neutrality result and its key implications for the Krugman (1980) model of trade extend from CES preferences to the entire class described by (1).12 Second, an increase of the spendable income of consumers has more complex implications. Consider the realistic case where higher income makes demand more rigid (h0 [ 0). Then, a positive demand shock reduces the perceived demand elasticity and induces firms to increase their markups and reduce sales accordingly. For a given number of firms this leads to a large positive impact on gross profits which promotes business creation and increases more than proportionally the number of varieties provided in the market.
Third, consider an increase in firms’ productivity associated with a reduction of the marginal cost (still assuming h0 [ 0): lower costs are translated less than proportion-ally to prices, which increases the markups and triggers additional entry (while the impact on firms size is ambiguous).13Accordingly, and contrary to what happens with CES preferences, our more general model allows demand and supply shocks to generate additional processes of business creation/destruction. This would alter the dynamic path of macroeconomic models with endogenous entry (for instance Bilbiie et al., 2012).
Under monopolistic competition and exogenous productivity, indirectly additive preferences exclude any impact of the number of goods on markups, which instead emerges under general additive or quadratic utilities (Melitz and Ottaviano, 2008; Zhelobodko et al., 2012). However, it is important to stress that such an impact is not really due to changes in the ‘intensity’ of competition between firms (where there are no strategic interactions) but to changes in the elasticity of substitution between goods perceived by consumers. Only concentrated markets with a small number of firms competing a la Bertrand or a la Cournot produce a direct impact of the number of firm on markups and our model can be easily extended in this direction to generate competitive effects (see Bertoletti and Etro (2014b) for a discussion of these effects under imperfect competition).14Alternatively, a link between market size and prices could be also introduced through endogenous productivity due to the adoption of specialised input, as in Murata (2009).
1.1. Examples
The results of our model can be illustrated with simple specifications that deliver closed-form solutions. Tractable cases arise if v01ðÞ is homogenous or logarithmic up to a linear transformation (see Behrens and Murata (2007) for a discussion under
12 As an immediate consequence, increasing the population just induces pure gains from variety. This is a
remarkable difference compared to the D–S model, where the existence of gains from trade can be guaranteed only when the equilibrium price is decreasing in the population (Zhelobodko et al., 2012; Dhingra and Morrow, 2015).
13 One can compute@ ln qe=@ ln c ¼ ðh 1Þðh fÞ=ð2h fÞ, therefore qe increases with the marginal
cost if and only if h> f: i.e., production increases with marginal cost if demand is log-concave. On the recent revival of the literature on incidence and pass-through, see for instance Fabinger and Weyl (2014). On empirical evidence on incomplete pass-through see De Loecker et al. (2012).
14 Competitive effects due to strategic interactions are widely studied in industrial organisation. For trade
implications see Bertoletti and Epifani (2014) and Etro (2015). For macroeconomic implications, see Etro and Colciago (2010).
direct additivity). This is clearly the case of the isoelastic function but also of the ‘addilog’ function vðsÞ ¼ ða sÞ1þc=ð1 þ cÞ with a > 0, which is the choke-off price, and c > 0, and of the exponential function vðsÞ ¼ ebs with b > 0.15
Let us consider the example based on the addilog preferences, which delivers a classic set of demand functions qi ¼ const ða pi=EÞ
c
whose elasticity depends on the parameter c: in particular, when c = 1 the demand is linear, when c?0 the demand is perfectly rigid and when c?∞ it is perfectly elastic. The addilog model leads to the following closed form solutions:
pe¼cc þ aE 1 þ c ; n e¼ðaE cÞEL F ðaE þ ccÞ; q e¼F ð1 þ cÞ aE c : (7)
The second example, based on exponential preferences, generates another widely used demand function with a log-linear specification, log qi ¼ const bpi=E, where b
affects the semi-elasticity of demand. The associated free-entry equilibrium can be easily derived as:16
pe¼ c þE b; n e¼ E2L F ðbc þ EÞ; q e¼Fb E : (8)
As expected, population is neutral on prices and firm’ size. Moreover, both examples satisfy h0[ 0, therefore higher income makes demand more rigid, which leads firms to increase their prices and reduce their production, with a more than proportional increase in the number of firms. In addition, a marginal cost reduction is not fully translated on prices, which attracts more business creation and has a limited impact on firm size (none with log-linear demand).
At this point, one may wonder what kind of direct utility functions are associated with indirect additivity and, in particular, with the above examples.17We can answer this question by solving for the inverse demand functions and then substituting them into (1) to recover the direct utility. The Roy identity (2) provides piðxi; E; lÞ ¼
Ev01ðlxiÞ for each variety i. Employing the budget constraint, we obtain that l is
implicitly defined by 1 ¼ Rjv01ðlxjÞxjdj. In the addilog example, we have:
piðxi; E; lÞ ¼ E a ðxij jÞl 1 c h i where l ¼ a Rn 0 xjdj 1 Rn 0 x 1þc c j dj 0 B @ 1 C A c : and for the log-linear demand we have:
piðxi; E; lÞ ¼ E b ln b l j j ln xi where l ¼ b exp b þ Rn 0 xjln xjdj Rn 0 xjdj ! ;
15 Other simple examples of v(s) are generalisations of the isoelastic function such as vðsÞ ¼ ðs þ g Þ1#,
withϑ > 1, or ‘mixtures’ such as vðsÞ ¼ s1aþ s1#withϑ 6¼ a > 1.
16 It is worth noticing that this case differs from the case of Logit demands (Anderson et al., 2012), which
derives from quasilinear preferences and exhibits no income effects.
17 Standard results ensure that, under our assumptions, preferences represented by (1) can be also
We can derive the direct utility for the general case as follows: U ¼ Z n 0 v v01ðlxjÞ dj Z n 0 uð lx Þdj with 1 ¼j Z n 0 v01ðlxjÞxjdj; (9)
where the ‘subutility’ u for each good is increasing in its consumption level. As expected, preferences are not directly separable: (9) shows that the marginal rate of substitution between two varieties is affected by the consumption of all the others through l. In our two examples we obtain:
U ¼ a Rn 0 xjdj 1 1þc ð1 þ cÞ Rn 0 x 1þc c j dj c and U ¼ Z n 0 xjdj exp b þRR0nxjln xjdj n 0 xjdj ! ;
respectively for the addilog and log-linear case, where utility depends on total consumption and on other aggregators of the consumption levels.18
1.2. Comparison with Other Models
To clarify the role of the assumptions on preferences in monopolistic competition, it is important to understand that demand elasticity in a symmetric equilibrium is ultimately determined by the elasticity of substitution.19In particular, indirect additivity amounts to assume that the optimal consumption ratio of any two goods does not depend on the price of any other good. This implies that in case of a common price pi ¼ pjthe elasticity
of substitution between varieties i and j does not depend on the number of goods, while it might depend on income. Instead, under direct additivity of preferences (the D–S model), it is the marginal rate of substitution between any two goods u0ðxiÞ=u0ðxjÞ which
is independent from the consumption of the other goods, leading to the property that their inverse price ratio, pi=pj, is independent from the quantities of the other goods
consumed. As an implication, the elasticity of substitution between varieties i and j in the case of a common consumption level xi ¼ xj depends only on this consumption level.
Finally, monopolistic competition under homothetic preferences (a case studied in Benassy, 1996) implies that the elasticity of substitution cannot depend on the level of income nor, in a symmetric equilibrium, on the price level. However, it may still depend on the number of varieties (as in the Translog example of Feenstra, 2003).
The different implications of these three classes of preferences are a consequence of their differences in the relevant demand elasticities and in the entry process. In case of direct additivity (D–S, 1977),20
income affects prices in the short run but not in the
18 Another example generating closed form solutions arises if vðsÞ ¼ ðs þ g Þ1#, withϑ > 1. Here the
equilibrium price is pe ¼ ð#c þ gEÞ=ð# 1Þ and the direct utility can be written as U ¼
½Rjxjð#1Þ=#dj#ð1 þ gRjxjdjÞ1#. Notice that h0? 0 if g ? 0.
19 The elasticity of substitution between goods i and j is a logarithmic derivative of x
i=xjwith respect to
pi=pj: see Blackorby and Russell (1989) for a formal discussion of the concept and Bertoletti and Etro (2014b)
for its crucial role in models of imperfect competition and product differentiation. See Mrazova and Neary (2013) for a article that investigates how the assumptions on demand determine the relevant comparative statics properties of market equilibria.
20 Under direct additivity the indirect utility function depends on a price aggregator given by the marginal
utility of income k: namely, V ¼R0nu½u01ðkpjÞdj with k implicitly defined by E ¼
Rn 0u
01ðkp jÞpjdj.
long run (Behrens and Murata, 2012a; Bertoletti and Etro, 2014a). To see this, notice that the reciprocal of the inverse demand elasticity, eðxiÞ ¼ u0ðxiÞ=u00ðxiÞxi, depends
on the consumption level only; the free-entry equilibrium can be summarised as follows: pe c pe ¼ 1 eðqe=LÞ; n e ¼ EL F eðqe=LÞ and q e¼F eðq½ e=LÞ 1 c ; (10)
where the equilibrium price does depend on the population L, which in turn affects non-linearly the number and the size of firms: the exact impact depends on whetherɛ is increasing or decreasing in consumption (Zhelobodko et al., 2012; Bertoletti and Epifani, 2014). However, the price and the production of each firm are determined independently from income E: in spite of non-homotheticity free entry neutralises the impact of income and markups cannot be affected by changes in consumer spending over the business cycle if the number of firms adjusts to ensure the zero profit condition.21
With homothetic preferences (Benassy, 1996), the demand elasticity in the case of symmetric consumption ɛ(n) can only depend on the number of varieties offered,22 and the free-entry equilibrium can be summarised as follows:
pe c pe ¼ 1 eðneÞ ; n e¼ EL F eðneÞ and q e¼F eðn½ eÞ 1 c : (11)
In this case, the number of firms increases more or less than proportionally with total income EL depending on whether ɛ(n) is increasing or decreasing in the number of firms, but is independent from the marginal cost, whose changes are (inversely) proportional to the firm size. Accordingly, markup is now neutral to productivity shocks (complete pass-through).
Summing up, under free entry, each one of the previous three classes of preferences is characterised by a different form of ‘neutrality’ and CES preferences inherit all of them:
REMARK 1. Under endogenous entry, with indirect additivity the population is neutral on
markups but income and productivity affect them; with direct additivity income is neutral on markups but population and productivity affect them; with homotheticity productivity is neutral on markups but population and income affect them.
1.3. Non-separable Indirect Utilities
As the reader might expect, it is possible to extend the setting of monopolistic competition to the general class of non-separable symmetric preferences: for a general treatment and new examples see Bertoletti and Etro (2014b).23The relevant indirect
21 The reason of the different free-entry results of direct and indirect additivity is rooted in the market
adjustment process, which takes place through shifts of demand due to changes in the mass of firms affecting the marginal utility of income. Since the profit expression with direct additivity is p ¼ ½u0ðxÞ=k cLx F , there is a unique (symmetric) equilibrium (zero-profit) value of k ¼ ½nu0ðxÞx=E. On the contrary, under indirect additivity, there is a unique equilibrium value of L=l ¼ LE= nv½ 0ðp=EÞp.
22 Homotheticity implies that the indirect utility function can be written V= Ek, where the marginal utility
of income k is a homogenous of degree 1 function of all prices.
23 In the case of a finite number of goods, Bertoletti and Etro (2014b) compare equilibria under
utility functions will in general depend on the number of the consumed varieties (as in Feenstra, 2003). The Roy identity provides always the relevant demand of each firm i as a function of its price pi and of some symmetric price aggregators whose values are
given for each firm (so that strategic considerations are neglected). In a symmetric equilibrium demand elasticity can be shown to be equal to the Morishima Elasticity of Substitution (defined in Blackorby and Russell, 1989), which depends either on the common price-income ratio p/E or on the common consumption value x, and on the number of varieties n (all related by the budget condition xpn = E). Accordingly, the equilibrium markup can be affected by all the exogenous parameters E, L/F and c. From this perspective, the three types of preferences discussed in the previous subsection represent the polar cases in which the equilibrium elasticity depends just either on p/E, x or n.
Here we limit our discussion to emphasise that the results obtained under indirect additivity extend also to other interesting non-separable preferences. Consider, for example, preferences that can be represented by the following ‘quadratic’ (non-homothetic) indirect utility function:
Vn¼1 2 Z n 0 a pj E 2 dj 1 2n Z n 0 pj E dj 2 : (12)
By the Roy identity we obtain the demand function:
xi¼
aE ðpi pÞ
l
j jE ; (13)
where l ¼ Eð@Vn=@EÞ \ 0, the additional price aggregator p ¼ Rn
0 pjdj=n is the
average price.24 Accordingly, a symmetric equilibrium yields the demand elasticity e ¼ pe=aE and the equilibrium price:
pe¼ c þ aE; (14)
which is increasing in income but again independent from the number of consumers, with firm size qe ¼ F =aE and number of firms ne ¼ aE2L=F ðc þ aEÞ.
2. Extensions
In this Section we extend our baseline model in a number of directions to emphasise its tractability and to investigate new issues.
2.1. Outside Good and Optimum Product Diversity
It can be useful to introduce an outside good representing the rest of the economy, as in many general equilibrium models.25 Let us consider a second sector producing a homogenous good under perfect competition and constant returns to scale. We follow
24 The demand presented in (13) is similar to that used to introduce monopolistic competition in the
textbook of Krugman et al. (2012, ch. 8). Income is irrelevant there since its microfoundation is inspired to the quasilinear model of Melitz and Ottaviano (2008).
25 A relevant part of the literature, starting with Spence (1976), has focused on quasilinear preferences,
D–S (1977, Section II) and adopt an indirect utility that has an intersectoral Cobb– Douglas form:26 V ¼ E p0 x Z n 0 v pj E dj 1x ; (15)
where p0is the price of the outside good and x 2 [0,1): clearly (15) collapses to (1) for
x = 0.27
In the Appendix, we show that the pricing rule for the differentiated goods remains the same as in (5), but the equilibrium mass of firms also depends on the elasticity of the indirect sub-utility v, defined as gðsÞ v0ðsÞs=vðsÞ [ 0, which reflects the relative importance of the differentiated sector:
PROPOSITION 3. In a Cobb–Douglas two-sector economy with indirect additivity and
monopolistic competition with endogenous entry in the differentiated sector, an increase in the number of consumers is neutral on prices and increases linearly the mass of firms but higher income increases prices if and only if the demand elasticity is increasing in the price.
It is interesting to evaluate the welfare properties of this generalised setting. As is well known, firms do not fully internalise the welfare impact of their entry decision, which may lead to too many or too few firms.28The constrained optimal allocation (controlling prices and number of varieties under a zero profit constraint) is derived in the Appendix and provides a simple comparison with the decentralised equilibrium for any x 2 [0,1):
PROPOSITION4. In a Cobb–Douglas two-sector economy with indirect additivity, monopolistic
competition with endogenous entry generates excess entry (insufficient entry) with too little (too much) production by each firm if the elasticity of the indirect sub-utility is everywhere increasing (decreasing) in the price.
Paralleling D–S, an intuition for this result can be obtained by noticing that g(s) approximates the ratio between the revenue of each firm and the additional utility generated by its variety. If g0[ ð\Þ0 they diverge and at the margin each firm finds it more profitable to price higher (lower), i.e., to produce less (more), than what would be socially desirable. This, in turn, attracts too many (too few) firms. One may find more reasonable the case in which the elasticity of the sub-utility decreases when income gets higher, which requires g0 [ 0. This is the case for the exponential and addilog cases: accordingly, they both imply excess entry.
26 A general specification of intersectoral preferences would not change the pricing rule, but the
equilibrium number of firms would not necessarily be linear in the number of consumers.
27 We can offer two interpretation of this setting. First, it corresponds to a typical two-sector trade model in
general equilibrium. Second it can be re-interpreted in terms of a two-period model where young agents have income E to be spent in the homogenous good or saved to consume the differentiated goods when old (with discount factor 1/x 1 and a zero interest rate). Such a set-up can be easily introduced in an overlapping generations (OLG) model.
28 See the original D–S (1977) paper, Kuhn and Vives (1999) and Dhingra and Morrow (2015) for key
references on this issue. Notice that the first-best allocation would require marginal cost pricing and subsidies to the firms.
2.2. Heterogeneous Consumers and Income Distribution
In this subsection, we generalise our model to the case of consumers with different preferences and income. The model remains tractable and allows one to draw implications on the impact of income distribution on the market structure (which is not neutral as in the CES case). We assume that there is a mass L of consumers of different ‘types’. Types are distributed across the population according to the cumulative distribution function C(h) with support [0,1].29 The consumer of type h has income Eh and indirect utility function given by:
Vh¼ Z n 0 vh pj Eh dj: (16)
As we prove in the Appendix, in the symmetric equilibrium, each firm adopts a simple extension of the pricing rule (5) for homogenous consumers:
pe c pe ¼ 1 ~hðpe; CÞ with ~hðp; CÞ Z 1 0 hh p Eh Eh E dCðhÞ; (17)
where ~h is a weighted average of the individual demand elasticities hhand the weight is
the consumer of type h’s ‘fraction’ of average income E. Under free entry, the mass L of consumer is again neutral but the distribution of types is not.30Moreover, we can prove additional results on the impact of income distribution on the market structure:
PROPOSITION 5. Under indirect additivity with heterogeneous consumers and monopolistic
competition with endogenous entry, an increase in the mass of consumers is neutral on prices and increases linearly the number of firms. With identical preferences:
(i) a change in income distribution according to the likelihood-ratio dominance raises (decreases) prices and increases the mass of firms more (less) than proportionally to average income if the demand elasticity is increasing (decreasing); and
(ii) a mean preserving spread decreases prices and the mass of firms if and only if the demand elasticity is convex.
The long-run impact of a change in the income distribution is in line with the spirit of the baseline model and breaks the neutrality emerging in models based on CES and exponential direct utilities or other equivalent microfoundations (Behrens and Murata, 2012b; Tarasov, 2014). However, the impact of a change in inequality is ambiguous. To fix ideas, let us consider the case of identical preferences with a demand elasticity increasing and convex with respect to the price, as in our addilog example: in such a case a mean preserving spread of the income distribution increases the average demand elasticity that is expected by the firms, which in turn reduces prices and induces business destruction.
29 We arrange consumer types in such a way that h> k implies E
h[ Ek, exclude any form of price
discrimination (i.e., there is no market segmentation) and focus on the symmetric equilibrium. For an analysis of heterogeneity in income (not in preferences) under direct additivity see Foellmi and Zweimuller (2004).
30 A special case arises if preferences are of the exponential type, i.e., v
h ¼ ebhp=Eh. In such a case
hhðp=EhÞ ¼ bhp=Ehand therefore ~h ¼ p b= E, where b ¼
R
hbhdCðtÞ: the market structure depends only on b
2.3. Heterogeneous Firms and Endogenous Quality
Melitz (2003) has shown that under heterogeneous productivity of the firms and CES preferences there are no selection effects on the set of active firms when markets expand, for instance in a boom or when the country opens up to costless trade. However, under direct additivity this neutrality holds for changes in income but not in the population, whose increase can give raise to ambiguous effects (depending on the shape of the elasticity of substitution). In particular, when prices are increasing with the size of consumption, an expansion of the market scale induces a selection effect, forcing the exit of the least productive firms, while less productive firms are able to survive during a contraction of the market (Zhelobodko et al., 2012; Bertoletti and Epifani, 2014). In this subsection we show that under indirect additivity and firms heterogeneity the market size remains neutral but income growth matters, exerting a sort of Darwinian effect as long as h0[ 0: income expansions allow less productive firms to survive but downturns lead to the exit of the least productive firms.
Following Melitz (2003), we assume that, upon paying a fixed entry cost Fe, each firm
draws its marginal cost c 2 [c,∞) from a continuous cumulative distribution G(c) with c ≥ 0. In the Appendix, we show that the equilibrium price function p(c) of an active c-firm satisfies the pricing rule (5), that high-productivity firms produce more, get larger revenues and are more profitable, as in Melitz (2003), but they also charge higher markups if and only if h0[ 0. Firms are active if their variable profits pv cover
the fixed cost F, that is if they have a marginal cost below the cut-off^c satisfying:31 pvð^cÞ ¼½pð^cÞ ^cvl0½pð^cÞ=EL¼ F : (18) Moreover, the equilibrium must satisfy the endogenous entry condition:
Z ^c
c
pvðcÞ F
½ dGðcÞ ¼ Fe; (19)
i.e., firms must expect zero profit from entering in the market. The two equations determine ^c and l as a function of L, F, Fe and E but in the Appendix we show that a
change in L produces no selection effects: an increase of market size is completely neutral on all the prices and on the productivity cut-off beyond which firms are active, even when preferences are not CES. Instead, changes in income induce novel effects on the structure of production:
PROPOSITION 6. Under indirect additivity, monopolistic competition with endogenous entry
and cost heterogeneity between firms, an increase in population is neutral on prices and on the productivities of the active firms (it increases proportionally their mass) but higher income increases prices of all firms and makes less productive firms able to survive if and only if the demand elasticity is increasing in the price.
31 Notice that even when the fixed costs of production are null (F= 0) but there is a finite choke-off price
s (as for instance in the case of addilog preferences) the marginal cost cut-off ^c ¼ sE is increasing in income and independent from population.
This result rationalises a Darwinian effect of recessions: these induce the exit of low-productivity firms, while on the contrary expansionary shocks associated with higher spending make low-productivity firms able to survive. Notice that such a cyclical process cannot be reproduced in the baseline Melitz model or in its extension to directly additive preferences.32
The heterogeneous costs model can be easily extended to take into account endogenous quality choices. This possibility has been recently explored to account for positive correlations of productivity with both quality and prices (see for instance Fajgelbaum et al. (2011) and Kugler and Verhoogen (2012)), whereby non-homothetic preferences are essential to explain the positive association of income with both quality and prices associated with the Linder hypothesis (Linder, 1961). Let us suppose that for a variety j with price pj and quality kj 0 the sub-utility is given by
vj ¼ vðpj=EÞuðkjÞ, where φ, u0[ 0 (higher quality increases both utility and demand
without affecting demand elasticity) and limk!0uðkÞ ¼ 0 to avoid corner solutions.
For simplicity, let us assume that a c-firm can produce goods of quality k at the marginal cost ck, obtaining variable profits pv ¼ ðp ckÞv0ðp=EÞuðkÞL=l. Under some
regular-ity conditions,33the equilibrium choices p(c) and k(c) satisfy the FOCs: p ck p ¼ 1 hðp=EÞ and h p E ¼ 1 þ ðkÞ; (20)
whereðkÞ u0ðkÞk=uðkÞ is the elasticity of demand with respect to quality. Price and quality are again independent from L34 but their relation with productivity and consumers’ income is more complex and can be derived through total differentiation as follows:
sign @p @c
¼ sign f g0 and sign @p
@E ¼ sign hf g;0 sign @k @c
¼ sign hf g0 and sign @k
@E
¼ sign hf g:0
First, notice that under CES preferences (h0 ¼ 0) quality is (endogenously) independent from productivity and consumers’ income, while when demand is isoelastic in quality (0 ¼ 0) the price is the same for all firms and more productive
firms invest more in quality. Under our standard assumption h0[ 0, more productive firms produce goods of higher quality. Moreover, they can even invest so much to sell them at higher prices compared to low productivity firms: this happens when demand becomes more sensitive to quality for products of higher quality (0 [ 0).35
Finally, again in line with the Linder hypothesis, higher income induces specialisation in high quality, high price goods. Notice that, given price and quality choices, variable profits are still increasing in productivity and income and the free-entry mechanism operates as above.
32 See Ottaviano (2012) for a related result in a model with a ‘quadratic’ direct utility and income effects. 33 The SOCs require 2h> f, n ku00=u0\ 2ðh 1Þ and ξ(f 2h) > (h 1)(2f 3h). Accordingly, it
must be the case that h0[ 0 if 0 0.
34 The neutrality of market size on quality is robust to more general specifications of the indirect sub-utility
as vj ¼ vðsj; kjÞ but, once again, it breaks down with direct additivity.
3. A Two-country Story
One of the main limits of the trade models based on monopolistic competition with CES preferences (Krugman, 1980; Melitz, 2003) is their inability in providing simple reasons why firms change markups when selling in different countries or facing different trade costs. It is well known that pricing to market is a pervasive phenomenon: identical products tend to be sold at different markups in different countries and, in particular, prices appear to be positively correlated with per capita income (Alessandria and Kaboski, 2011) but hardly with country population (Hand-bury and Weinstein, 2014; Simonovska, 2015). We have also evidence of incomplete pass-through of cost reductions due to trade liberalisation (De Loecker et al., 2012). In this Section, we generalise the basic Krugman (1980) model to indirectly additive preferences and emphasise its implications for the structure of trade.36
We consider trade between two countries sharing the same preferences (1) and technology, as embedded into the costs c and F, but possibly with different numbers of consumers L and L and different productivity (i.e., labour endowment in efficiency units). In particular, we assume that the labour endowments of consumers in the home and foreign countries are respectively e and e, so that income levels are E = we and E ¼ we. Accordingly, the marginal and fixed costs in the domestic and foreign countries are respectively wc and wF and wc and wF .37Let us assume that to export each firm bears an ‘iceberg’ cost s ≥ 1 and, as standard, let us rule out the possibility of parallel imports aimed at arbitraging away price differentials (i.e., international markets are segmented). Consider the profit of a firm i, based in the home country, which can choose two different prices for domestic sales pi and exports pzi:
pi¼ ðpi wcÞv0 pi E L l þ ðpzi swcÞv0 pzi E L l wF ; (21)
where l and l are the home and foreign values of (3). A symmetric expression holds for a foreign firm j, choosing prices pj and pzj.
The optimal price rules for the home firms are: p wc p ¼ 1 h p E ; pz swc pz ¼ 1 h pz E ; (22)
and the price rules for the Foreign firms are similarly obtained. Therefore, four different prices emerge in the symmetric equilibrium, with l ¼ nv0ðp=EÞp=E þ nv0ðpz=EÞpz=E
36 Several recent articles have studied trade in multi-country models with non-homothetic preferences.
Behrens and Murata (2012a) and Simonovska (2015) use specific types of directly additive preferences: the former article assumes that markets are not segmented while the latter deals with the case of international price discrimination. Mrazova and Neary (2014), Bertoletti and Epifani (2014) and Bertoletti and Etro (2014a) assume direct additivity: the first two articles consider identical countries (with the former focusing on welfare with symmetric firms and the latter dealing with the case of heterogeneous firms), while the third one investigates the case of different countries and market segmentation. Fajgelbaum et al. (2011) consider products of different qualities within a Logit demand system.
37 Identical results would emerge assuming different productivities (affecting proportionally both
marginal and fixed cost) and equal labour endowments. This would be reflected on the equilibrium wages and through this on incomes.
and l¼ nv0ðpz=EÞpz=Eþ nv0ðp=EÞp=E. The endogenous entry condition for the
firms of the home country reads as:
ðp wcÞv0 p E L l þ ðpz swcÞv0 pz E L l ¼ wF ; (23)
and a corresponding one holds for the firms of the foreign country. We can normalise the home wage to unity, w = 1, and close the model with the domestic resource constraint (or, equivalently, the labour market clearing condition):
eL ¼ n cxL þ scxð zLþ FÞ; (24)
where x ¼ v0ðp=EÞ=l and xz ¼ v0ðpz=EÞ=l. This provides a system of seven equations
in seven unknowns (p, pz, p, pz, w, n and n). With non-homothetic preferences,
population and productivity of a country have a distinct impact on the relative wages, with complex implications for price differentials and the structure of trade. However, we can get the main insights focusing on the two cases traditionally analysed in the literature: costless trade between different countries, and costly trade between identical countries. The former case is obtained setting s = 1 and is characterised as follows:
PROPOSITION 7. Under indirect additivity, monopolistic competition with endogenous entry
and costless trade, firms adopt a higher price in the country with higher per capita income; opening up to trade reduces (increases) the number of firms in the country with higher (lower) per capita income if and only if the demand elasticity is increasing, and generates pure gains from variety.
Since costless trade induces wage equalisation (wages must be the same in both countries otherwise the zero-profit conditions could not be simultaneously satisfied; also see Behrens and Murata, 2012a), the price rules show immediately the emergence of pricing to market: under the standard assumption h0 [ 0, prices of identical goods are higher in the country with the higher per capita income because demand is more rigid compared to the other country and these prices are independent of the population sizes.38Consumers enjoy new varieties produced abroad and bought at the same price of the domestic goods. Nevertheless, opening up to trade induces a redistribution of firms and production across countries which is absent in the Krugman (1980) model. Firms exporting to the country with poorer consumers sell there at a lower mark up and face entry of foreign firms in the domestic market: accordingly, they obtain lower variable profits, which leads to business destruction at home. Therefore, the richer country is characterised by a process of concentration in fewer and larger firms. On the contrary, business creation takes place in the poorer country, where firms start selling abroad at higher mark ups and reduce their size. Finally, prices and the total number of firms across countries remain the same as in autarky, therefore the gains from trade are always pure gains from variety as in Krugman (1980).39
38 Notice that pricing to market can arise in a multi-country setting also under direct additivity (Markusen,
2013; Bertoletti and Etro, 2014a; Simonovska, 2015): a larger income implies a larger individual consumption for each variety, which in turn affects markups. However, exactly for this reason, direct additivity also preserves the (ambiguous) impact of the number of consumers on markups.
39 Augmenting the model with strategic interactions generates a genuine competitive effect of trade on
The second case we consider, the one of costly trade, is obtained by setting s > 1 with L ¼ L and e ¼ e and is characterised as follows:
PROPOSITION 8. Under indirect additivity, monopolistic competition with endogenous entry
and costly trade between identical countries, opening up to trade reduces the markup on the exported goods and the mass of firms in each country relative to autarky if and only if the demand elasticity is increasing.
Because also trade costs are symmetric, wages and prices are equalised in both countries. However, the markup applied to goods sold at home and abroad is not the same when preferences are not homothetic. In particular, the markup (on the marginal cost c s) is lower for the exported goods if h0 [ 0, because firms undershift transport costs on prices (De Loecker et al. (2012) provide convincing evidence for such incomplete pass-through of the cost reductions on prices during trade liberalisations). This shows a different form of pricing to market, which has the additional consequence of affecting the entry process compared to the neutrality of the Krugman (1980) model: as long as the average markup diminishes because of undershifting of the transport costs on export prices, opening up generates a process of business destruction in both countries. Welfare gains from liberalisation, therefore, cannot be guaranteed, since the changes in the price of the imported goods also affects the gains from variety. In particular, trade liberalisation (a reduction of s) reduces the price of imports and increases their consumption but also affects the number of firms through changes in the average markup, which leads to an ambiguous impact on welfare. We prove the following in the Appendix:40
REMARK2. Trade liberalisation has a non-monotonic welfare impact when h0[ 0, since it is
negative around autarky (assuming the existence of a finite choke-off price) and it must be positive in the case of an opening to free trade.
Finally, under some additional conditions, we can show that richer countries trade relatively more between themselves than poorer countries, which is also in line with the evidence (for instance see Fieler, 2011). The export share on GDP, say - xzpz=ðxzpz þ xpÞ, can be derived as:
- ¼ v0ðpz=EÞpz
v0ðpz=EÞpzþ v0ðp=EÞp
(25)
where prices satisfy (22). Under CES preferences this ratio is 1=ð1 þ sh1Þ, therefore
the export share is independent from income. Under non-homotheticity, weak additional conditions satisfied with linear and log-linear demand (f0 [ 0 is sufficient) guarantee that the export share increases with income because the relative demand for imported goods becomes more elastic. This is in line again with the Linder hypothesis that richer countries trade more between themselves.
In conclusion, we note that the assumption of indirect additivity of preferences may prove useful also to disentangle the impacts of income and population on other
aspects of trade such as the volume of trade between countries, the emergence of multinationals and multiproduct firms and the role of trade policy.
4. Conclusion
The contribution of this work is to propose a new tractable model of monopolistic competition. We study the equilibria of large markets under indirect additivity of consumers’ preferences, an alternative to the classic assumptions of direct additivity or quasilinearity. Under reasonable conditions (namely more rigid demand for higher income), indirect additivity generates simple predictions that are in contrast with the traditional approach and await for empirical tests: higher income (and productivity) in a market should increase markups and more than proportionally the number of firms, while the number of consumers should be neutral. Our framework encompasses a number of convenient cases such as those with addilog and exponential preferences generating linear and log-linear demand functions. Therefore it could be applied to analyse complex issues usually considered exclusive territory for CES modelling: in particular, we hope that indirect additivity will prove useful in building new multi-country trade models and dynamic macroeconomic models with monopolistic competition and heterogeneous firms and consumers.
Appendix. Proofs
Proof of Proposition 1. By using h0p=hE ¼ h þ 1 f ? 0 if and only if h þ 1 ? f, the result
follows from the total differentiation of (5): @ ln pe @ ln n ¼ 0; @ ln pe @ ln E ¼ h þ 1 f 2h f and @ ln pe @ ln c ¼ 1 h þ 1 f 2h f ; (A.1)
after noticing that 2h f> 0 from the SOC.
Proof of Proposition 2. Using the comparative statics in (A.1) and differentiating (6) we obtain: @ ln ne @ ln L ¼ 1; @ ln ne @ ln E ¼ 1 þ ðh þ 1 fÞðh 1Þ 2h f and @ ln ne @ ln c ¼ ðh þ 1 fÞðh 1Þ 2h f :
Proof of Proposition 3. By the Roy identity, the demand of each differentiated good is given by: xi¼ v0ðpi=EÞ Z n 0 v0ðpj=EÞ pj E x 1 xvðpj=EÞ dj ; and the profits of each firm i are given by:
pi¼ v0ðpi=EÞðpi cÞL Z n 0 v0ðpj=EÞpj=E x 1 xvðpj=EÞ h i dj F ; (A.2)
where the denominator is unaffected by a single price. It is immediate to verify that, independently from the value of x, each firm adopts the same pricing rule as in (5) and the
comparative static properties of the profit-maximising price pe are then the same as in
Proposition 1. The number of goods produced in the free-entry equilibrium can be derived as:
ne¼ ELð1 xÞ g ðp
e=EÞ
F hðpe=EÞ ð1 xÞ g ðp½ e=EÞ þ x ; (A.3)
which now depends on x and both the elasticities h and g: changes in income affect prices and the allocation of expenditure between the differentiated goods and the outside one. Nevertheless, the number of firms is always proportional to L.
Proof of Proposition 4. We compare the market equilibrium with a constrained optimal allocation which maximises utility under a zero-profit condition for the firms. The problem boils down to: max n;p nvðp=EÞ s.t. p ¼ c þ F xðp; n; EÞL; where xðp; n; EÞ ¼ v0ðp=EÞ n v0ðp=EÞp E x 1 xvðp=EÞ h i
is the symmetric demand of a variety. Notice that the zero-profit constraint implictly defines n as
a function of p that is continuous on ½c; sE, with n(c) = 0 and limp!sEnðpÞ which is a finite
number. Accordingly, the objective function is null for p= c and p ! sE, and positive for at least
some intermediate price due to our assumptions. Therefore, an internal optimum satisfying the FOCs must exist:
vðp=EÞ ¼ q F Lx2 @x @n; nv0ðp=EÞ=E ¼ q 1 þ F Lx2 @x @p ; where q is the relevant Lagrange multiplier. The FOCs imply:
gðp=EÞ ¼ xpL F þ @ ln xðp; n; EÞ @ ln p @ ln xðp; n; EÞ @ ln n ¼ p p cþ @ ln x ðp; n; EÞ @ ln p ;
and it is easily computed that:
g0¼g 1 h þ g½ p=E : (A.4) Since @ ln xðp; n; EÞ @ ln p ¼ xd ln gðp=EÞ d ln p=E x þ ð1 xÞg 1 ¼ ð2x 1Þ g xh x þ ð1 xÞ g ; we obtain the optimal markup:
p c
p ¼
x þ ð1 xÞ g
Finally, the optimal mass of firms is:
n¼ ð1 xÞ gLE
F ð1 xÞ gð1 þ gÞ þ xh½ : (A.6)
Comparing (A.5) with (5), it follows that the RHS of (A.5) is larger (smaller) than 1/h
if (everywhere) h? 1 + g. Since from (A.4) it follows that g07 0 is equivalent to 1/(1 + g) ? 1/h,
then (everywhere) g07 0 is equivalent to pe7 p, which in turn implies x7 xe by the zero-profit
constraint. Using the fact that the RHS of (A.6) is larger (smaller) than the RHS of (A.3) if g0is
smaller (larger) than zero we obtain ne7 nif g07 0, which completes the proof.
Proof of Proposition 5. The demand of a consumer h for good i can be written as xhiðpi; Eh; lhÞ ¼ vh0ðpi=EhÞ=lh, where lh ¼
R
jv 0
hðpj=EhÞðpj=EhÞdj. Since types are distributed
according to C(h) with support [0,1], profits of firm i are:
pðpiÞ ¼ ðpi cÞL
Z 1
0
xhiðpi; Eh; lhÞdCðhÞ F ;
which implies that the profit-maximising price pisatisfies the FOC:
ðpi cÞ Z 1 0 v00h pi Eh lhEh dC ðhÞ þ Z 1 0 vh0 pi Eh lh dC ðhÞ ¼ 0:
Symmetric pricing implies lh ¼ nvh0ðpe=EhÞðpe=EhÞ and thus:
pe c pe ¼ Z 1 0 Eh npedCðhÞ Z 1 0 vh00 p e Eh nvh0ðpe=E hÞ dCðhÞ ¼Z 1 1 0 hh p e Eh Eh EdCðhÞ 1 ~hðpe; CÞ;
where hh vh00ðp=EhÞp=½vh0ðp=EhÞEh and E ¼
R
hEhdCðhÞ. The price rule is thus independent
from L. Endogenous entry implies the following mass of firms:
ne ¼ EL
F ~h ðpe; CÞ;
which proves the first part of the proposition since L affects linearly ne.
To prove the second part, suppose that all consumers share the same preferences. It is then
convenient to rewrite ~h directly as ~hðp; I Þ ¼ RE1
E0 hðp=EÞðE= E ÞdI ðE Þ, where I() is the income
distribution function implied by C(). To prove (1), consider a change in I according to
likelihood ratio dominance: i.e., a change from I0to I1such that i1ðEÞ=i1ðT Þ i0ðEÞ=i0ðT Þ for
all E> T, E; T 2 ½E0; E1, where iðÞ ¼ I0ðÞ is the relevant density function. This implies that I1
also (first-order) stochastically dominates I0: thus, by a well-known result, this raises the average
income (i.e., E1[ E0). We can write:
~hðpe; I Þ ¼ Z E1 E0 h pe E
dUðE; I Þ with UðE; I Þ ¼
Z E
E0 T
EdI ðT Þ;
where the cumulative distribution functionΦ has density U0. Notice that:
U0ðE; I1Þ U0ðT ; I1Þ¼ Ei1ðEÞ Ti1ðT Þ Ei0ðEÞ Ti0ðT Þ¼ U 0ðE; I0Þ U0ðT ; I0Þ for all E[ T ; E; T 2 E½ 0; E1:
Accordingly,UðE; I1Þ dominates in terms of the likelihood ratio UðE; I0Þ and it must then be the
case that UðE; I1Þ UðE; I0Þ for all E 2 E
0; E1
½ (i.e., the former distribution first-order
stochastically dominates the latter). It follows that when h(p/E) is a decreasing (increasing) function of E an improvement of income distribution according to likelihood-ratio dominance
implies ~hðp; I1Þ ð Þ~hðp; I0Þ for all p, which in turn decreases (increases) the equilibrium
value of ~h and thus raises (decreases) the equilibrium price level and the mass of active firms
more (less) than proportionally to the rise of average income.
To prove (2), suppose that I1 is a mean-preserving spread of I0. Then E1 ¼ E0 ¼ E . The
function hðp=EÞE= E in the definition of ~hðp; I Þ is a concave (convex) function with respect to E
if and only if h00\ð [ Þ0. By a standard result, it must then be the case that ~hðp; I0Þ [ ð\Þ~hðp; I1Þ
when h00\ð [ Þ0. It follows that a mean-preserving spread decreases (increases) the equilibrium
value of ~h, and then raises (decreases) prices and the mass of firms when h is a concave (convex)
function of the price.
Proof of Proposition 6. The variable profits of an active c-firm are given by
pv ¼ ðp cÞv0ðp=EÞL=l, where l is independent of its price choice. Therefore, the pricing
rule (5) applies to all firms. We denote with p= p(c) the profit-maximising price of a c-firm, with
xðcÞ ¼ v0½pðcÞ=E=l the individual consumption of its product, and with:
pvðcÞ ¼ pðcÞ c½ v0½pðcÞ=EL=l
its variable profit for given l defined as: l ¼ n
Z ^c c
v0½pðcÞ=E½pðcÞ=EdGðcÞGð^cÞ \0:
Note that the optimal price of firm c does not depend upon L and l and follows the same
comparative statics as in Proposition 1, with@ lnp(c)/@ lnc 71 and @ lnp(c)/@ lnE 7 0 if and
only if h0? 0. Moreover, the FOCs and SOCs for profit maximisation imply the following
elasticities with respect to the marginal cost: @ ln xðcÞ @ ln c ¼ h½pðcÞ=E @ ln pðcÞ @ ln c \0; (A.7) @ ln pðcÞxðcÞ @ ln c ¼ fh½pðcÞ=E 1g2
2h½pðcÞ=E f½pðcÞ=E\0; (A.8)
@ ln pvðcÞ
@ ln c ¼ 1 h½pðcÞ=E\0: (A.9)
Accordingly, high-productivity (low-c) firms are larger, make more revenues, are more profitable and they charge higher (lower) markups if h is increasing (decreasing). In addition, for the given l, we have the following elasticities with respect to income:
@ ln xðcÞ
@ ln E ¼
h½pðcÞ=E2 h½pðcÞ=E
2h½pðcÞ=E f½pðcÞ=E[ 0; (A.10)
@ ln pðcÞxðcÞ
@ ln E ¼
h½pðcÞ=E2þ 1 f½pðcÞ=E
2h½pðcÞ=E f½pðcÞ=E [ 1; (A.11)
@ ln pvðcÞ
@ ln E ¼ h½pðcÞ=E [ 1: (A.12)
The size of each firm increases with E (for given l), and revenues and profits increase more than
proportionally. However, each price increases with respect to income only when h0[ 0 and
positive profits. Denote by ^c the marginal cost cutoff, namely the value of c satisfying the zero
cutoff profit condition pvð^cÞ ¼ F , or:
pð^cÞ ^c
½ v0½pð^cÞ=EL ¼ lF : (A.13)
The relation (A.13) implicitly defines^c ¼ ^cðE; lj jF=LÞ. Differentiating it yields:
@ ln ^c @ ln E¼ h½pð^cÞ=E h½pð^cÞ=E 1[ 0; (A.14) @ ln ^c @ ln lj j¼ @ ln^c @ ln F ¼ @ ln^c @ ln L¼ 1 1 h½pð^cÞ=E\0: (A.15)
Endogenous entry of firms in the market implies that expected profits: Efpg ¼
Z ^c c
pvðcÞ F
½ dGðcÞ; (A.16)
must be equal to the sunk entry cost Fe. The expected profits decrease when the absolute value of
l increases, that is @E{p}/@|l| < 0. Accordingly, the condition Efpg ¼ Fe pins down uniquely
the equilibrium value of l as a function lðE; L; F ; FeÞ. Moreover, using (A.13) the free entry
condition can be written as: Z ^c c pðcÞ c ½ v0½pðcÞ=E pð^cÞ ^c ½ v0½pð^cÞ=E 1 dGðcÞ ¼Fe F : (A.17)
The system{(A.13), (A.17)} can actually be seen as determining ^c and l in function of L, F, Fe
and E. The second equation, (A.17), fixes ^c and is independent from L, while the first one,
(A.13), determines l as linear with respect to L. The cut-off^c is therefore independent of market
scale, because l proportionally adjusts in such a way to keep constant the ratio L/l and thus the variable profit of the cut-off firm. As a consequence, as in Melitz (2003), a change in L produces no selection effect, even when preferences are not CES. However, (A.17) shows that changes in the fixed costs are not neutral on the set of active firms: an increase in the fixed cost of
production F or a reduction in the fixed cost of entry Fe require a reduction in the equilibrium
value of the cut-off ^c, which correspond to the typical selection effect a la Melitz (2003). The
impact of income E is instead more complex. Since by (A.12) and (A.14) an increase of E raises E{p}, it must increase the equilibrium value of |l| to satisfy zero expected profits. In particular, we have: @ ln lj j @ ln E ¼ @E p½ @E @E p½ @ lj j E l j j¼ hð^cÞ [ 0; where (A.18) hð^cÞ ¼ Z ^c c 1 h½pðcÞ=E pðcÞxðcÞ R^c c pðcÞxðcÞdG ðcÞ dGðcÞ 8 < : 9 = ; 1
is the harmonic mean of the h values according to G() and^c. Computing the total derivative of ^c
with respect to E using (A.14), (A.15) and (A.18), we obtain: d ln^c d ln E¼ @^ c @Eþ @^ c @ lj j @ lj j @E E ^c¼ h½pð^cÞ=E hð^cÞ h½pð^cÞ=E 1 ; (A.19)
which is positive if and only if (everywhere) h0[ 0: that is, in addition to increasing (decreasing)
the mark up of the inframarginal firms, a rise of E creates an anti-selection (selection) effect if h increases (decreases) with respect to the price. To close the model, the expected mass of active firms
